# Periodicity of Hyperbolic Cosine

## Theorem

Let $k \in \Z$.

Then:

$\cosh \left({x + 2 k \pi i}\right) = \cosh x$

## Proof

 $\displaystyle \cosh \left({x + 2 k \pi i}\right)$ $=$ $\displaystyle \frac {e^{x + 2 k \pi i} + e^{- \left({x + 2 k \pi i}\right)} } {2 i}$ Definition of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac {e^{i \left({-i x + 2 k \pi}\right)} + e^{i \left({i x + 2 \left({-k}\right) \pi}\right)} } {2 i}$ $i^2 = -1$ and simplifying $\displaystyle$ $=$ $\displaystyle \frac {e^{i \left({-i x}\right)} + e^{i \left({i x}\right)} } {2 i}$ Periodicity of Complex Exponential Function $\displaystyle$ $=$ $\displaystyle \frac {e^x + e^{- x} } {2 i}$ $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle \cosh x$ Definition of Hyperbolic Cosine

$\blacksquare$